Mathematical modeling of nanomachining with bimodal dynamic scanning thermal microscope probe

Abstract

The aim of current study is to present and study the bimodal dynamic Scanning Thermal Microscope (SThM) for nanomachining. Bimodal SThM is the same as conventional SThM where the base is excited simultaneously at a frequency close to the 1st and 2nd resonance frequency of probe. Bimodal excitation makes two separated channels for monitoring sample data for different purposes. Thus, a mathematical model for bimodal SThM is provided. The nonlinear coupled equations of motion are solved analytically. The presented model is compared with the literature and the behavior of bimodal SThM is compared with conventional type. It is shown that the shifted resonance frequencies are independent of the type of excitation and bimodal excitation increases amplitudes at all shifted resonance frequencies. By decreasing ratio of base excitation amplitudes, the amplitude in the 2nd base excitation frequency and all shifted resonance frequencies increase. Bimodal technique amplifies the amplitudes at the 2nd shifted resonance frequency, this data can improve resolution in thermal images. Then, the effects of tip radius on the final surface are simulated. It is shown that the nanomachining depth increases significantly for bimodal SThM and nanomachined surface roughness is increased. It is declared increase in temperature gradient does not necessarily improve the roughness and the final surface always depends on the interactions of three parameters: the tip radius, the total shape of the vibrational response and the location of the peaks with large amplitudes.

Appendix

$q_{l}$ is obtained from the solving equations by the Kramer method:

$$ q_{1} = \frac{{\left| {\begin{array}{*{20}c} {F_{1}^{1} \cos \omega_{\rm e}^{1} t + F_{1}^{2} \cos \omega_{\rm e}^{2} t} & {K_{12} } & {K_{13} } \\ {F_{2}^{1} \cos \omega_{\rm e}^{1} t + F_{2}^{2} \cos \omega_{\rm e}^{2} t} & {M_{2} D^{2} + K_{22} } & {K_{23} } \\ {F_{3}^{1} \cos \omega_{\rm e}^{1} t + F_{3}^{2} \cos \omega_{\rm e}^{2} t} & {K_{32} } & {M_{3} D^{2} + K_{33} } \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {M_{1} D^{2} + K_{11} } & {K_{12} } & {K_{13} } \\ {K_{21} } & {M_{2} D^{2} + K_{22} } & {K_{23} } \\ {K_{31} } & {K_{32} } & {M_{3} D^{2} + K_{33} } \\ \end{array} } \right|}} $$

(37)

$$ q_{2} = \frac{{\left| {\begin{array}{*{20}c} {M_{1} D^{2} + K_{11} } & {F_{1}^{1} \cos \omega_{\rm e}^{1} t + F_{1}^{2} \cos \omega_{\rm e}^{2} t} & {K_{13} } \\ {K_{21} } & {F_{2}^{1} \cos \omega_{\rm e}^{1} t + F_{2}^{2} \cos \omega_{\rm e}^{2} t} & {K_{23} } \\ {K_{31} } & {F_{3}^{1} \cos \omega_{\rm e}^{1} t + F_{3}^{2} \cos \omega_{\rm e}^{2} t} & {M_{3} D^{2} + K_{33} } \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {M_{1} D^{2} + K_{11} } & {K_{12} } & {K_{13} } \\ {K_{21} } & {M_{2} D^{2} + K_{22} } & {K_{23} } \\ {K_{31} } & {K_{32} } & {M_{3} D^{2} + K_{33} } \\ \end{array} } \right|}} $$

(38)

$$ q_{3} = \frac{{\left| {\begin{array}{*{20}c} {M_{1} D^{2} + K_{11} } & {K_{12} } & {F_{1}^{1} \cos \omega_{\rm e}^{1} t + F_{1}^{2} \cos \omega_{\rm e}^{2} t} \\ {K_{21} } & {M_{2} D^{2} + K_{22} } & {F_{2}^{1} \cos \omega_{\rm e}^{1} t + F_{2}^{2} \cos \omega_{\rm e}^{2} t} \\ {K_{31} } & {K_{32} } & {F_{3}^{1} \cos \omega_{\rm e}^{1} t + F_{3}^{2} \cos \omega_{\rm e}^{2} t} \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {M_{1} D^{2} + K_{11} } & {K_{12} } & {K_{13} } \\ {K_{21} } & {M_{2} D^{2} + K_{22} } & {K_{23} } \\ {K_{31} } & {K_{32} } & {M_{3} D^{2} + K_{33} } \\ \end{array} } \right|}} $$

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